On some power series with algebraic coefficients and Liouville numbers

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On some power series with algebraic

coefficients and Liouville numbers

Gül Karadeniz Gözeri

*

*Correspondence: gulkaradeniz@gmail.com Department of Mathematics, Faculty of Science, Istanbul University, Vezneciler, Istanbul, Turkiye

Abstract

In this work, we consider some power series with algebraic coefficients from a certain algebraic number fieldKof degreemand investigate transcendence of the values of the given series for some Liouville number arguments.

1 Introduction

The theory of transcendental numbers has a long history and was originated back to Li-ouville in his famous paper [] in which he produced the first explicit examples of tran-scendental numbers at a time where their existence was not yet known. Later, Cantor [] gave another proof of the existence of transcendental numbers by establishing the denu-merability of the set of algebraic numbers. It follows from this that almost all real numbers are transcendental. Further, the theory of transcendental numbers is closely related to the study of Diophantine approximation. Recent advances in Diophantine approximation can be found in the excellent surveys of Moshchevitin [] and Waldschimidt [].

Mahler [] introduced a classification of the set of all transcendental numbers into three disjoint classes, termedS,T andU and this classification has proved to be of consider-able value in the general development of the subject. The first classification of this kind was outlined by Maillet in [], and others were described by Perna in [] and Morduchai-Boltovskoj [] but to Mahler’s classification attaches by for the most interest. Mahler de-scribed this classification in the following way.

LetP(x) =anxn+· · ·+ax+abe a polynomial with integral coefficients. The height H(P) ofPis defined byH(P) =max(|an|, . . . ,|a|) and the degree ofPis denoted bydeg(P). Given an arbitrary complex numberξ, Mahler puts

ωn(H,ξ) =minP(ξ):deg(P)≤n,H(P)≤H,P(ξ)= , wherenandHare positive integers. Next Mahler puts

ωn(ξ) =H→∞lim

–logωn(H,ξ)

logH and ω(ξ) =n→∞lim ωn(ξ)

n .

The inequalities ≤ωn(ξ)≤ ∞and ≤ω(ξ)≤ ∞hold. Fromωn+(H,ξ)≤ωn(H,ξ), we getωn+(ξ)≥ωn(ξ). If for an indexωn(ξ) =∞, theμ(ξ) is defined as the smallest of them, otherwiseμ(ξ) =∞. Thus,μ(ξ) is uniquely determined. Furthermore, the two quantities μ(ξ) andω(ξ) are never finite simultaneously, for the finiteness ofμ(ξ) implies that there

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is an n<∞such thatωn=∞, whenceω=∞. Therefore, there are the following four possibilities forξ,ξis called

anA- number ifω(ξ) = ,μ(ξ) =∞, anS- number if  <ω(ξ) <∞,μ(ξ) =∞, aT- number ifω(ξ) =∞,μ(ξ) =∞, aU- number ifω(ξ) =∞,μ(ξ) <∞.

In [], Koksma introduced an analogous classification of complex numbers. He divided the complex numbers into four classesA∗,S∗,T∗andU∗in the following way.

Letαbe an arbitrary algebraic number. If we denote its minimal defining polynomial by P(x), then the heightH(α) ofα is defined byH(α) =H(P) and the degreedeg(α) ofαis defined bydeg(α) =deg(P). Given an arbitrary complex numberξ and positive integersn, H, letαbe an algebraic number with degree at mostnand height at mostHsuch that|ξα| takes the smallest positive value; Koksma definesωn(H,ξ) by the following equation:

ωn(H,ξ) =min|ξα|:αis algebraic,deg(α)≤n,H(α)≤H,α=ξ. Next, Koksma puts

ωn(ξ) = lim

H→∞

–log(n(H,ξ))

logH and ω

(ξ) = lim n→∞

ωn(ξ)

n .

The inequalities ≤ωn(ξ)≤ ∞and ≤ω∗(ξ)≤ ∞hold. If for an indexωn(ξ) =∞, the μ∗(ξ) is defined as the smallest of them, otherwiseμ∗(ξ) =∞. Thus,μ∗(ξ) is uniquely determined. Furthermore, the two quantitiesμ∗(ξ) andω∗(ξ) are never finite simultane-ously. Therefore, there are the following four possibilities forξ,ξis called:

anA∗- number ifω∗(ξ) = ,μ∗(ξ) =∞, anS∗- number if  <ω∗(ξ) <∞,μ∗(ξ) =∞, aT∗- number ifω∗(ξ) =∞,μ∗(ξ) =∞, aU∗- number ifω∗(ξ) =∞,μ∗(ξ) <∞.

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sums. In [], Nyblom employed a variation on the proof used to established Liouville’s theorem concerning the rational approximation of algebraic numbers, to deduce explicit growth conditions for a certain series to converge to a transcendental number. Later, Ny-blom [] derived a sufficiency condition for a series of positive rational terms to converge to a transcendental number. Further, Duverney [] proved a theorem that gives a crite-rion for the sums of infinite series to be transcendental. The terms of these series consist of the rational numbers and converge regularly and very quickly to zero. In [], Hančl introduced the concept of transcendental sequences and proved a criterion for sequences to be transcendental. Later, a new concept of a Liouville sequence was introduced in [] by means of the related Liouville series. Some recent results for the transcendence of infi-nite series can also be found in Borwein and Coons [], Hančl and Rucki [], Hančl and Štěpnička [], Murty and Weatherby [], Weatherby [].

In the present work, we considered certain power series with algebraic coefficients from a certain algebraic number fieldKof degreemand showed that under certain conditions these series take values belonging to either the algebraic number fieldK ormi=Ui in Mahler’s classification of the complex numbers for some Liouville number arguments.

2 Preliminaries

In this paper, |x| means the absolute value of x and the least common multiple of x,x, . . . ,xnis denoted by [x,x, . . . ,xn].

Definition  A real numberξis called a Liouville number if and only if for every positive integernthere exists integerspn,qn(qn> ) with

 <ξpn qn

< 

qn n .

The set of all Liouville numbers is identical with the U subclass. More information about Liouville numbers may be found in [–]. Now, in order to prove our main the-orem we need the following lemmas.

Lemma [] Letα, . . . ,αk (k≥)be algebraic numbers which belong to an algebraic number field K of degree m, and let F(y,x, . . . ,xk)be a polynomial with rational inte-gral coefficients and with degree at leastin y. Ifηis any algebraic number such that F(η,α, . . . ,αk) = ,thendeg(η)≤dm and

H(η)≤dm+(l+···+lk)mHmH(α)lm· · ·H(αk)lkm,

where H is the height of the polynomial F,d is the degree of F in y and liis the degree of F in xifor i= , . . . ,k.

Lemma [] Letαbe an algebraic number of degree m,and letα()=α, . . . ,α(m)be its conjugates.Then|α| ≤H(α),where|α|=max(|α()|, . . . ,|α(m)|).

Lemma [] Letα be an algebraic number of degree m,then H(α)≤(|α|)m,where

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3 Main result

Theorem  Let K be an algebraic number field of degree m,and let

g(x) = ∞

n=

αn en

xn

be a power series such thatαnK are non-zero algebraic numbers and en> are rational integers satisfying the following conditions:

lim

n→∞

logen+

logen

=η> , ()

lim

n→∞

logen+

logen

=∞, ()

lim

n→∞

logH(αn)

logen

=μ< . ()

Further,letξbe a Liouville number satisfying the following two properties:

. ξhas an approximation with rational numbers pnqn (qn> )so that the following

inequality holds for sufficiently largen

ξpn

qn

< 

qnsnn

lim

n→∞sn= +∞

. ()

. There exist two positive real constantsγandγwith ηη–<γ<γand

nqnne

γ

n ()

for sufficiently largen.

Then g(ξ)belongs to either the algebraic number field K ormi=Ui.

Proof It follows from () that

logen+>ηlogen ()

for sufficiently largen, whereη=ηεandεis to be chosen as  <ε<ηγγ– . It follows from () that the sequence{en}is strictly increasing, thus we have

lim

n→∞en=∞, ()

lim

n→∞

logen

n =∞ and n→∞lim

logen

n =∞. ()

Furthermore, from () we get

en<e

η

n+ ()

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LetEn= [e,e, . . . ,en]. Then by using () and (), we obtain for sufficiently largen

enEne

ε+ η

η–

n , ()

whereεis to be chosen as  <ε<γ–η–η . We can easily deduce from () andμ< μ+ that

logH(αn)

logen

<μ+ 

 ()

for sufficiently largen. Since () holds, there is a natural numberNsuch that

H(αn) <e(

μ+  )

n ()

for everyn>N. On the other hand, we get from Lemma  that

|αn| ≤H(αn) ()

sinceαnare algebraic numbers. From here and (), we obtain

|αn| ≤ |αn| ≤e (μ+ )

n ()

for everyn>N. Now, we shall define the algebraic numbers

βn= n

γ=

αγ

pn qn

γ

forn= , , , . . . . SinceβnK,deg(βn)≤mforn= , , , . . . . Let us determine an up-per bound for the heights of the algebraic numbersβn. By multiplying both sides of this equality byEnqnnand puttingli=Enei fori= , , , . . . , we obtain the equality

Enqnnβn=

lqnnα+

lqnn–pn

α+· · ·+

lnpnn

αn.

Sinceξ is a Liouville number, we can assume thatpn=  forn= , , , . . . . Then we get a polynomial

G(y,x,x, . . . ,xn) =Enqnnyn

γ=

lγqnnγpγn

with rational integral coefficients such thatG(βn,α, . . . ,αn) = . Further, this polynomial is of degree  in eachy,x,x, . . . ,xn. Thus, we deduce from Lemma  that

H(βn)≤m+(n+)mHmH(α)m· · ·H(αn)m, () whereHis the height of the polynomialG(y,x,x, . . . ,xn). By using (), we obtain|pnqn|ikn

fori= , , , . . . , wherek=|ξ|+  >  is a real constant. From here, we can easily get

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It follows from () and () that

H(βn)kmnEmnqmnn H(α)m· · ·H(αn)m, ()

wherek= k>  is a real constant. Moreover, we get

H(βi)≤

|βi|

m

(i= , , , . . .) ()

from Lemma . Then we deduce from () and () that

H(βn)≤m(n+)kmnEmnqmnn |β| · · · |βn|

m

. ()

By using (), () and (), we obtain

|β| · · · |βn|

m

≤m(n+)keγm(μ+

 )

n ,

wherek=max(, (H(α)· · ·H(αN))

m) is a real constant. It follows from here, () and () that

H(βn)knek

n , ()

wherek= mkmk>  andk=γm+γm(μ+) +γm>  are real constants. Now, we consider the following polynomials:

gn(x) = n

ν=

αν

forn= , , . . . . Sincegn(x) are continuous and differentiable for all real numbers, at least one real numbercnexists betweenξ andpnqn such that for everyn

gn(ξ) –βn=gn(ξ) –gn pn qn

= ξpn qn

gn(cn). ()

It is obvious that|cn| ≤max(|ξ|,|pnqn|). Since|pnqn|<|ξ|+ , we obtain|cn| ≤ |ξ|+  for suffi-ciently largen. Furthermore, from here and () and (), we get

gn(ξ) –gn pn qn

≤ 

qnnsn n

ν=

|αν|

ν|ξ|+ ν– ()

for sufficiently largen.

Defineσn=max(|α|, . . . ,|αn|). Then we obtain n

ν=

|αν|

ν|ξ|+ ν–≤nσn

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for sufficiently largen. It follows from () thatσn≤e (+μ)

n for sufficiently largen. We get from here and (), ()

gn(ξ) –βn

n(|ξ|+ )n–e(+μ) n qnsnn

.

By using (), we obtain from here that

gn(ξ) –βn

n(|ξ|+ )n– sn–(

+μ

 ) n

. ()

From () andlimn→∞sn=∞, it is possible to find a sequence{ωn}withlimn→∞ωn=∞ such that

n(|ξ|+ )n– sn–(

+μ

 ) n

≤ 

(knekn)ωn

. ()

Therefore, we get from (), () and ()

gn(ξ) –βn≤ 

H(βn)ωn ()

for sufficiently largen. Moreover, the following inequality holds:

g(ξ) –gn(ξ)≤ ∞

i=

|αn+i| en+i |

ξ|n+i.

We get from here and ()

g(ξ) –gn(ξ)≤|ξ| n+

e( –μ

 ) n+

 + en+ en+

–μ

|ξ|+ en+ en+

–μ

|ξ|+· · ·

.

Thus, we deduce from () that

 < en+ en+

<  e(– 

η) n+

.

Since () holds, then we obtainlimn→∞enen++= . Similarly, since –μ> , we have

en+ en++k

–μ

|ξ|k<  

k

fork= , , , . . . and, therefore,

g(ξ) –gn(ξ)≤|ξ| n+

e( –μ

 ) n+

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On the other hand from (), we get |ξ|n+e(–μ)

n+ for sufficiently largen. From here, we obtain

g(ξ) –gn(ξ)≤ 

e( –μ

 ) n+

for sufficiently largen. Now if we definern=loglogenen+, then we have

g(ξ) –gn(ξ)≤ 

ern( –μ

 ) n

.

Using (), then it follows that there exists a subsequence{rnk}of{rn}such thatlimk→∞rnk=

∞. Therefore, we get

g(ξ) –gnk(ξ)≤ 

ernk( –μ

 ) nk

()

for sufficiently largenk. From () andlimk→∞rnk=∞, there exists a suitable sequence{rnk} withlimk→∞rnk=∞such that

ernk( –μ

 ) nk

≤ 

(kne

kn)rnk

and, therefore, from () and (), we obtain

g(ξ) –gnk(ξ)≤ 

H(βnk)rnk

()

for sufficiently largenk. On the other hand by using (), we deduce that

gnk(ξ) –βnk≤ 

H(βnk)ωnk

()

for sufficiently largenk. Letωnk=min(rnk,ωnk). It follows from () and () that

g(ξ) –βnk≤  H(βnk)ωnk

,

wherelimk→∞ωnk=∞. It follows from here thatlimk→∞βnk=g(ξ). Thus, if the sequence

{βnk}is constant, theng(ξ) is an algebraic number inK. Otherwiseg(ξ)∈

m

i=Ui.

4 Conclusion

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Competing interests

The author declares that she has no competing interests.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The author would like to express her sincere thanks to the referees for their careful reading and for making some valuable comments, which have essentially improved the presentation of this paper.

Received: 14 December 2012 Accepted: 2 April 2013 Published: 16 April 2013

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doi:10.1186/1029-242X-2013-178

Cite this article as:Karadeniz Gözeri:On some power series with algebraic coefficients and Liouville numbers.

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